The current-modified nonlinear Schrödinger equation

Abstract

By comparison with both experimental and numerical data, Dysthe's (1979) O(ε4) modified nonlinear Schrödinger; equation has been shown to model the evolution of a slowly varying wavetrain well (here ε is the wave steepness). In this work, we extend the equation to include a prescribed, large-scale, O(ε2) surface current which varies about a mean value. As an introduction, a heuristic derivation of the O(ε3) current-modified equation, used by Bakhanov et al. (1996), is given, before a more formal approach is used to derive the O(ε4) equation. Numerical solutions of the new equations are compared in one horizontal dimension with those from a fully nonlinear solver for velocity potential in the specific case of a sinusoidal surface current, such as may be due to an underlying internal wave. The comparisons are encouraging, especially for the O(ε4) equation.

Probability distributions of surface gravity waves during spectral changes

“…Trulsen & Dysthe 1996) that only allows for a limited time horizon, τ , for the calculations of the order of (ω p s 3 ) −1 , where ω p = 2π/T p . Quantitative comparisons with wave tank experiments and other numerical wave models (Lo & Mei 1985;Trulsen & Stansberg 2001;Shemer et al 2001Shemer et al , 2002Stocker & Peregrine 1999;Clamond & Grue 2002) have indicated that a reasonable time horizon is indeed τ ≈ (ω p s 3 ) −1 , which for the present simulations (with…”

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

“…Note that a term of that kind arises in the Eulerian description of potential wave oscillations of the free surface as well. In the paper by Stocker and Peregrine (1999), U = U * sin (kx − ω t) was chosen and interpreted as a harmonically changing surface current induced by an internal wave. We shall further take U = 0.…”

Trapping and instability of directional gravity waves in localized water currents

“…Higher-order expansions in terms of nonlinearity and wave dispersion have been used to derive more general Schrödinger-like models for water waves including currents [22]. In the formalism of Ref.…”

“…In the formalism of Ref. [22], the current velocity is of order 2 , while the time and space derivatives and Ak 0 are of order , where is a nonlinearity parameter. Since the dispersion and nonlinearity coefficients D x and −ξ have equal signs, Eq.…”